1 Introduction

With the continuous development of coal resources in China, shallow resources are becoming scarcer and resources with a buried depth of 1000 m are now under consideration. With the increase of mining depth, deep coal masses are subject to a complex stress environment that includes high ground stress, high earth temperature, high karst water pressure, and mining disturbance (Gao et al. 2021; Xie et al. 2019). Coal has high elastic potential energy, which increases the possibility of dynamic disasters, such as coal and gas outbursts and rock bursts, when subjected to external dynamic failure. For this reason, most studies have focused on dynamic mechanical characteristics of the deep surrounding rock. In the present study, dynamic constitutive models of coal at different depths were explored by considering the high ground stress of deep-occurrence conditions. This research determined the infinitesimal element strength of coal by considering the rate effect and improves the dynamic constitutive model applicable to different depths, providing a theoretical reference for coal engineering phenomena at different depths.

Present dynamic constitutive models of coal can be mainly divided into element- and strength-type methods (Wang et al. 2016). The element method regards elasticity, plasticity, and viscosity as three basic properties of coal, and considers that the macroscopic mechanical properties of rock materials reflect the comprehensive performance of these three properties: a spring element is used to simulate a perfect elastic body that conforms to Hooke’s law; a friction plate (or slide block) element is used to simulate a perfect plastic body; and a piston damper element with holes is used to simulate a viscous body that conforms to Newton’s fluid law. These three basic elements are combined to form the differential constitutive equation of the loaded rock, and the correlation coefficients are experimentally determined. In terms of a dynamic constitutive model of the element type, Lindholm et al. (1974) proposed an excess stress model in 1974 in which stress was determined by strain and strain rates. In the 1980s and 1990s, Kinoshita et al. (1977) and Yu (1989) proved and developed this model, and proposed a modified excess stress model that reflects characteristics of rock strength that change with strain rate, but cannot reflect the rate effect of elastic modulus. Zheng and Xia (1996) regarded rock as a viscoelastic body, and established a viscoelastic continuous damage constitutive model by considering damage changes in the deformation process. Xie et al. (2013) studied the dynamic deformation process of rock by combining damage theory, viscoplasticity theory, and the excess stress model, and proposed a new dynamic damage constitutive model. Zhai et al. (2011) combined damage evolution with the element model theory to establish a viscoelastic–plastic dynamic constitutive model of rock by considering the damage. Shan et al. (2006) combined the statistical damage model with the viscoelastic model, regarded rock events as the conjoint of a damaged body and sticky cylinder, and proposed a dynamic time-dependent damage model. Li et al. (2006a, b) adopted the viscoelastic damage element model to establish a combined dynamic and static loading constitutive model at moderate strain rate. Fu et al. (2013) conducted dynamic impact tests on anthracite, analyzed experimental curve characteristics, and established a damage–viscoelastic model. Xu and Liu (2014) introduced a statistically damaged body into the existing viscoelastic constitutive model and proposed a dynamic statistically damaged constitutive model for marble that could reflect the effect of high temperature. The element model reflects the dynamic mechanical behavior of coal well but has many disadvantages, such as too many parameters, unclear physical meaning of the parameters, and the need for manual trial calculations (Wang et al. 2016).

The strength-type method establishes the strength statistical damage constitutive model by combining classical rock mechanics theories of strength with statistical damage theory. Krajcinovic and Silva (1982) first proposed a statistical damage theory in 1982, based on macroscopic phenomenology. This approach did not pursue microscopic damage mechanisms but only paid attention to the impact of infinitesimal element body damage on macroscopic properties. Its process did not differ from that of general damage theory and it was convenient to use. On the basis of the randomness of defect distribution in rock materials, a statistical damage constitutive model was constructed by combining the theories of statistical strength and continuum damage, introducing a new approach for study of the rock-damage softening constitutive model. Statistical damage theory based on macroscopic phenomenology has many advantages, such as fewer parameters, convenient application, and the ability to reflect the final failure of rock materials due to damage, and thus it has been widely applied to the static damage constitutive model of coal (Wang et al. 2007; Deng and Gu 2011). Wang et al. (2016) proposed a strength-type dynamic statistical damage constitutive model of coal by theoretical analysis based on dynamic stress–strain curves of coal under different impact loads and compared this with the element-type statistical damage constitutive model, finding that the correlation coefficient of the former was higher and in good agreement with the test results. Cao et al. (2017) improved existing dynamic strength criteria of rock, established nonlinear rock dynamic strength criteria that reflect the influence of strain rate, and introduced statistical damage theory to establish a rock dynamic statistical damage constitutive model that showed good consistency with dynamic triaxial test curves. Zheng et al. (2021) used a split Hopkinson pressure bar system with a diameter of 50 mm to carry out uniaxial impact compression tests on coal at 20–100 s−1 dynamic strain rate, and established a dynamic strength-type statistical damage constitutive model of coal based on Weibull statistical distribution and Drucker–Prager failure criteria. Zhai et al. (2022) tested dynamic compression properties of sandstone specimens under different freeze–thaw cycles using an SHPB system with a diameter of 50 mm, and proposed a dynamic damage constitutive model that considers various forms of dissipated energy. Kong et al. (2021) carried out dynamic compression experiments on gas-bearing coal based on the improved SHPB system, established a dynamic constitutive equation for this coal under impact load, and verified it by comparing theoretical and experimental results.

In summary, previous scholars have conducted in-depth research on the dynamic mechanical test and constitutive model establishment of rock. Taking coal samples as the research object, the previous research mainly focused on the dynamic mechanical test and constitutive model establishment of coal under uniaxial stress conditions. The establishment of dynamic constitutive model of coals considering deep occurrence conditions is rarely reported. In this study, the dynamic mechanical tests of coals with different occurrence depths were carried out systematically by the improved separated SHPB test system with confining pressure. Based on the statistical damage theory, the expression for infinitesimal element strength of coal considering the rate effect was established, and the dynamic constitutive model of coals with different depths was further established and verified, and the calculation method of relevant parameters was determined.

2 Experimental

2.1 Sample preparation and test scheme

The test coal was taken from the No. 10 mine in the Pingdingshan mining area 24,130 rock protective layer working face (Henan province, China) (Gao et al. 2020). The effective strike length of this working face is 709 m, the inclined width is 157–160.5 m, and the dip angle of the coal seam is 5.9°–13.2°. A comprehensive histogram is shown in Fig. 1. According to the sample processing method recommended by the International Society of Rock Mechanics, a single bulk coal sample was divided into standard coal samples. To eliminate the influence of bedding and human disturbance on the sample properties, the direction of sample drilling was parallel to the direction of bedding. Sample preparation involved dry drilling, dry cutting, and dry grinding. To ensure flatness of the sample, both end faces were ground and polished to ensure that they were parallel within ≤ 0.02 mm. A processed sample is shown in Fig. 2.

Fig. 1
figure 1

Comprehensive histogram of working face

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Fig. 2
figure 2

Standard specimen after processing

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The loading wave adopted by the traditional SHPB experimental technique can be approximated as a square wave (Frew et al. 2002). In order to avoid the distortion of the experimental results, the method recommended by ISRM is adopted, and C1100 copper alloy disc (diameter: 12.75 mm; thickness: 0.7 mm) was added as a shaper material to improve the square loading wave relative to the bevel loading wave (Culshaw 2015). Comparison of the loading wave before and after shaping is shown in Fig. 3. An improved SHPB test system with confining pressure was used in these experiments, as shown in Fig. 4 (Wu et al. 2016; Xu et al. 2019).

Fig. 3
figure 3

Comparison of loading wave before and after shaping

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Fig. 4
figure 4

Improved split Hopkinson pressure bar system with confining pressure

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Based on the geological conditions on site, the average density of the overlying rock is determined to be 2500 kg/m3 (Li et al. 2023), and different depths are established to apply equivalent stress loads. To determine the dynamic strength of coal at different depths, the improved SHPB system was first used to load prepared coal samples to initial hydrostatic pressures of 0, 5, 10, 15, and 20 MPa to simulate original ground stress environments of the coal at depths of 0, 200, 400, 600, and 800 m, respectively. Dynamic mechanical testing of the coal at different impact rates under stress conditions at different depths was then carried out. Mechanical parameters, such as peak stress, peak strain, and dynamic strength of the coal under different loading rates, were analyzed according to the test results, and dynamic damage constitutive models of the coal at different depths were constructed and verified.

2.2 Determination of coal infinitesimal element strength considering rate effect

Coal has numerous original defects, such as micro fissures and joint structures. These original defects gradually propagate and coalesce under the action of external loads, until macroscopic fracture occurs. Damage mechanics is most commonly used to study damage evolution of rock materials. Determination of the infinitesimal element strength of coal is key to establishing the damage constitutive model. The damage constitutive model based on random distribution of infinitesimal element strength and the dynamic damage constitutive model based on the element model were analyzed, combined with comprehensive consideration of the properties of deep surrounding rock, dynamic deformation characteristics of the coal, and the stress environment at different depths, to determine the infinitesimal element strength of the coal considering the rate effect. Using the proposed dynamic strength criterion of coal considering the rate effect (Li et al. 2023), a method to measure the dynamic infinitesimal element strength of coal was established, as shown in Eq. 1:

$$F_{{\text{d}}} = \sigma^{\prime}_{1d} - \sigma^{\prime}_{3d} \frac{1 + \sin \varphi }{{1 - \sin \varphi }} - \frac{2c\cos \varphi }{{1 - \sin \varphi }}$$
(1)

where Fd is the dynamic infinitesimal element strength of coal, \(\dot{\sigma }\) is the loading rate, \(\dot{\sigma }_{0}\) is the reference loading rate, 1 GPa/s.

c is the cohesive forces, \(c = 0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5451}\) (Li 2019),

φ is the angle of internal friction, \(\varphi = 0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5360}\) (Li 2019),

\(\sigma^{\prime}_{{1{\text{d}}}}\), \(\sigma^{\prime}_{{3{\text{d}}}}\) is the dynamic effective stress of coal.

On the basis of Lemitre’s strain equivalence hypothesis, we assumed that damaged coal does not have bearing capacity; the entire load is borne by undamaged coal. Therefore, the constitutive relation of the damaged material only needs to replace stress in the constitutive relation of the original material by the effective stress, then:

$$\sigma_{id} = \sigma_{id}^{\prime } (1 - D_{d} )\quad (i = 1,2,3)$$
(2)

where \(\sigma_{id}\) is the dynamic macroscopic nominal stress of coal, \(\sigma_{2d} = \sigma_{3d} = \sigma_{3}\), and \(D_{d}\) is the dynamic damage variable of coal.

Assuming the stress–strain relationship of coal prior to damage obeys a linear elastic relationship, then:

$$\sigma_{1d} = E_{d} \varepsilon_{1d} \left( {1 - D_{d} } \right) + 2\mu \sigma_{3d}$$
(3)

where \(\varepsilon_{1d}\) is the axial strain, \(E_{d}\) is the dynamic elastic modulus of coal, and \(\mu\) is Poisson’s ratio of the coal.

The above relations can be substituted into Eq. (1) to determine a strong expression for the infinitesimal element of coal considering the rate effect:

$$\begin{aligned} F_{d} {\text{ }} & = \frac{{E_{d} \varepsilon _{{1d}} \left[ {\left( {\sigma _{{1d}} - \sigma _{{3d}} } \right) - \left( {\sigma _{{1d}} + \sigma _{{3d}} } \right)\sin \varphi } \right]}}{{\sigma _{{1d}} - 2\mu \sigma _{{3d}} }} - \frac{{2c\cos \varphi }}{{1 - \sin \varphi }}{\text{ }} \\ & = \frac{{E_{d} \varepsilon _{{1d}} \left[ {\left( {\sigma _{{1d}} - \sigma _{{3d}} } \right) - \left( {\sigma _{{1d}} + \sigma _{{3d}} } \right)\sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right]}}{{\left( {\sigma _{{1d}} - 2\mu \sigma _{{3d}} } \right)\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ & \quad - \frac{{2\left( {0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5451}} } \right)\cos \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}}{{\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ \end{aligned}$$
(4)

Mechanical parameters are obtained from dynamic tests that consider the occurrence depth of the coal, and thus this method to determine the dynamic infinitesimal element strength of coal is universal. Equation (4) applies to the determination of infinitesimal element strength in dynamic mechanical testing of coal at different depths.

3 Construction and verification of dynamic damage constitutive models of coal at different depths

3.1 Construction and parameter determination

According to the above determination of infinitesimal element strength, the dynamic damage constitutive model of coal at different occurrence depths was established by statistical theory. We assumed that coal comprises numerous infinitesimal elements under dynamic load, and the strength Fd of each infinitesimal element obeys a Weibull distribution. The probability density function is then:

$$P(F_{d} ) = \frac{m}{{F_{0} }} \cdot \left( {\frac{{F_{d} }}{{F_{0} }}} \right)^{m - 1} \cdot \exp \left[ { - \left( {\frac{{F_{d} }}{{F_{0} }}} \right)^{m} } \right]$$
(5)

where Fd is the dynamic infinitesimal element strength, and m and F0 are Weibull distribution parameters, which reflect mechanical properties of the coal.

The coal damage variables and statistical damage model are then determined:

$$D_{d} = \left\{ {\begin{array}{*{20}l} {\int_{0}^{{F_{d} }} {P(y)dy = 1 - \exp \left[ { - \left( {\frac{{F_{d} }}{{F_{0} }}} \right)^{m} } \right]} } \hfill & {(F_{d} \ge 0)} \hfill \\ 0 \hfill & {(F_{d} < 0)} \hfill \\ \end{array} } \right.$$
(6)
$$\sigma_{1d} = \left\{ {\begin{array}{*{20}l} {E_{{\text{d}}} \varepsilon_{1d} \exp \left[ { - (F_{d} \, /F_{0} )^{m} } \right] + 2\mu \sigma_{3d} } \hfill & {(F_{d} \ge 0)} \hfill \\ {E_{{\text{d}}} \varepsilon_{1d} + 2\mu \sigma_{3d} } \hfill & {(F_{d} < 0)} \hfill \\ \end{array} } \right.$$
(7)

From the extreme values of the triaxial stress–strain curve of coal:

$$\left. {\frac{{\partial \sigma_{1d} }}{{\partial \varepsilon_{1d} }}} \right|_{\begin{subarray}{l} \sigma_{1d} = \sigma_{1dp} \\ \varepsilon_{1d} = \varepsilon_{1dp} \end{subarray} } = 0$$
(8)

where \(\sigma_{1dp}\) and \(\varepsilon_{1dp}\) are the respective stress and strain values corresponding to the peak in the stress–strain curve.

The experimentally measured curves are those of deviatoric stress and deviatoric strain, and thus they must be converted before they can be substituted into the calculation equation. \(\sigma_{1dp}\) is the sum of the dynamic strength \(\sigma_{cd}\) of coal; hydrostatic pressure \(\sigma_{1s}\) is given by Eq. 9:

$$\sigma_{1dp} = \sigma_{cd} + \sigma_{1s}$$
(9)

Assuming that the corresponding peak point strain measured in dynamic testing of coal under hydrostatic pressure is \(\varepsilon^{\prime}_{1dp}\), \(\varepsilon_{1dp}\) is the sum of the peak strain \(\varepsilon^{\prime}_{1dp}\), and \(\varepsilon_{c}\) is the initial strain measured in the test, then:

$$\varepsilon_{1dp} = \varepsilon^{\prime}_{1dp} + \varepsilon_{c}$$
(10)

where \(\varepsilon_{c}\) is the initial strain of coal under the action of the hydrostatic pressure \(\sigma_{1s} = \sigma_{2s} = \sigma_{3s}\). This analysis assumes that the coal will not be damaged under the application of triaxial equi-compression hydrostatic pressure.

When no load is applied, the initial strain of coal under hydrostatic pressure is:

$$\varepsilon_{c} = {{(1 - 2\mu )\sigma_{3} } \mathord{\left/ {\vphantom {{(1 - 2\mu )\sigma_{3} } {E_{d} }}} \right. \kern-0pt} {E_{d} }}$$
(11)

Equations (9) and (10) are substituted into Eqs. (7) and (4) to obtain:

$$\sigma_{1dp} = E_{d} \varepsilon_{1dp} \exp [( - F_{dp} /F_{0} )^{m} ] + 2\mu \sigma_{3}$$
(12)
$$\begin{aligned} F_{{dp}} & = \frac{{E_{d} \varepsilon _{{1dp}} \left[ {(\sigma _{{1dp}} - \sigma _{3} ) - (\sigma _{{1dp}} + \sigma _{3} )\sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right]}}{{(\sigma _{{1dp}} - 2\mu \sigma _{3} )\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ & \quad - \frac{{2\left( {0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5451}} } \right)\cos \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}}{{\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ \end{aligned}$$
(13)

After solving Eq. (8), the Weibull distribution parameters m and F0 can be obtained in combination with Eq. (12) as follows:

$$m = - \frac{{F_{dp} (2\mu \sigma_{3} - \sigma_{1dp} )(1 - \sin \varphi )}}{{E_{d} \varepsilon_{1dp} [(\sigma_{1dp} - \sigma_{3} ) - (\sigma_{1dp} + \sigma_{3} )\sin \varphi ]\ln ((\sigma_{1dp} - 2\mu \sigma_{3} )/E\varepsilon_{1dp} )}}$$
(14)
$$F_{0} = \frac{{F_{dp} }}{{\left[ { - \ln \frac{{\sigma_{1dp} - 2\mu \sigma_{3} }}{{E_{d} \varepsilon_{1dp} }}} \right]^{1/m} }}$$
(15)

As shown in Fig. 5, with the increase of occurrence depth, the dynamic peak stress of coal shows an overall increasing trend, with an increase of 13.15%, 34.08%, and 66.62% in the average peak stress, respectively. However, the peak strain shows an overall decreasing trend, with a decrease of 11.73%, 19.30%, and 13.67% in the average peak strain, respectively. Due to the differences in mechanical characteristic parameters of coal with different depths. The parameters \(\sigma_{1dp}\) and \(\varepsilon_{1dp}\) and the formulas to calculate parameters m and F0 in the model are related to the occurrence depth. Therefore, relationships for \(\sigma_{1dp}\) and \(\varepsilon_{1dp}\) at different depths were established to obtain methods to determine parameters m and F0 that can be used to construct a unified damage statistical constitutive model of coal that reflects different occurrence depths.

Fig. 5
figure 5

Dynamic stress–strain curves of coal at depths of 200 m, 400 m, 600 m, and 800 m

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According to the dynamic Mohr–Coulomb criterion for coal at different depth stress conditions obtained from the above test results:

$$\begin{aligned} \sigma _{{1dp}} & = \frac{{2\sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}}{{1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}} \cdot 20\left( {\frac{H}{{H_{0} }}} \right) \\ & \quad + 2 \cdot \left( {0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5451}} } \right) \cdot \frac{{\cos \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}}{{1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}} \\ \end{aligned}$$
(16)

where H is the coal depth (m) and H0 is the characteristic depth (800 m).

The theoretical solution of the dynamic compressive strength at a typical loading rate at different depths can be obtained from Eq. (16), as shown in Table 1. Errors between the dynamic strength values calculated according to the dynamic Mohr–Coulomb criterion and the experimental strength values are less than 8.8770%, and the test results are well reflected by the model.

Table 1 Relationship between theoretical value and experimental strength of dynamic compressive strength of coal under different depth stress conditions
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Dynamic stress–strain curves of the coal at different depths are shown in Fig. 5. These show that the dynamic elastic modulus of coal does not change with loading rate for the same depth, while the dynamic elastic modulus of coal rock increases significantly with the increase of depth, and the maximum increase of the mean dynamic elastic modulus is 90.32%. The dynamic elastic modulus values of coal at different depths are shown in Table 2, and the relationship between occurrence depth and the mean value of dynamic elastic modulus is shown in Fig. 6.

Table 2 Dynamic mechanical characteristic values of coal at different depths
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Fig. 6
figure 6

Relationship between coal depth and dynamic elastic modulus

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The relationship between dynamic elastic modulus and depth can be established from Fig. 6 as follows:

$$E_{d} = 0.189\sigma_{3} + 2.3 = 3.76({H \mathord{\left/ {\vphantom {H {H_{0} ) + 2.3}}} \right. \kern-0pt} {H_{0} ) + 2.3}}$$
(17)

The linear correlation coefficient of the fitting equation R is 0.9877. The errors between the dynamic elastic moduli determined from the fitting equation and the test results were 4.52%, 2.87%, 4.49%, and 2.71%, respectively, for the four depths considered. The errors between the two are very small, and thus the fitting curve accurately reflects the test results.

Equation (10) shows that \(\varepsilon_{1dp}\) is the sum of strains \(\varepsilon^{\prime}_{1dp}\) at the peak point measured in a test and the initial strain \(\varepsilon_{c}\) of the sample under hydrostatic pressure. According to Eq. (11), the initial strains of coal were 0.0471, 0.0627, 0.0805, and 0.0954 when the depths were 200 m, 400 m, 600 m, and 800 m, respectively.

Simultaneous solution of Eqs. (10) and (11) can be obtained:

$$\varepsilon_{1dp} = \varepsilon^{\prime}_{1dp} + (1 - 2\mu_{d} ){{\sigma_{3} } \mathord{\left/ {\vphantom {{\sigma_{3} } {E_{d} }}} \right. \kern-0pt} {E_{d} }}$$
(18)

The peak strain relationship corresponding to different depths and loading rates of coal is shown in Fig. 7 according to the data presented in Table 2. The peak strain and loading rate present linear correlations for different coal depths. The fitting equations and correlation coefficients are shown in Table 3.

Fig. 7
figure 7

Peak strain relationship corresponding to different depths and loading rates of coal

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Table 3 Linear fitting equation and correlation coefficient
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Further fitting of the strain results corresponding to the peak stress at different depths can be obtained:

$$\varepsilon^{\prime}_{1dp} = A\dot{\sigma } + B(H/H_{0} ) + C$$
(19)

where A, B, and C are 0.001, − 40.72, and 0.733, respectively.

Substituting Eq. (19) into Eq. (18) yields:

$$\varepsilon_{1dp} = A\dot{\sigma } + B(H/H_{0} ) + C + (1 - 2\mu_{d} ){{f(H/H_{0} )} \mathord{\left/ {\vphantom {{f(H/H_{0} )} {E_{d} }}} \right. \kern-0pt} {E_{d} }}$$
(20)

The occurrence depth of deep surrounding rock significantly affects its mechanical properties and damage characteristics. On the basis of the established dynamic damage constitutive model of coal and considering the influence of stress environment at different depths, the constitutive model reflecting the damage characteristics of coal at different depths is as follows:

$$\sigma_{1d} = E_{d} \varepsilon_{1d} \exp [ - (F_{d} /F_{0} )^{m} ] + 2\mu f(H/H_{0} )$$
(21)

The Weibull distribution parameters m and F0 of dynamic strength infinitesimal elements of coal at different depths are determined as:

$$m = - \frac{{F_{dp} [2\mu f(H/H_{0} ) - \sigma_{1dp} ]}}{{E_{d} \varepsilon_{1dp} \left\{ {[(\sigma_{1dp} - f(H/H_{0} )] - \sigma_{1dp} + f(H/H_{0} )\sin \varphi } \right\}\ln \frac{{\sigma_{1dp} - 2\mu f(H/H_{0} )}}{{E_{d} \varepsilon_{1dp} }}}}$$
(22)
$$F_{0} = \frac{{F_{dp} }}{{\left[ { - \ln \frac{{\sigma_{1dp} - 2\mu f(H/H_{0} )}}{{E_{d} \varepsilon_{1dp} }}} \right]^{1/m} }}$$
(23)

The dynamic damage constitutive model of coal considering the occurrence depth then can be written as follows:

$$\begin{aligned} \sigma_{1d} & = E_{d} \varepsilon_{1d} \exp \left[ { - \left( {\left. {\frac{{E\varepsilon_{1d} [(\sigma_{1d} - f(H/H_{0} )) - (\sigma_{1d} + f(H/H_{0} ))\sin \varphi ] - 2c\cos \varphi [\sigma_{1d} - 2\mu f(H/H_{0} )]}}{{[(\sigma_{1d} - 2\mu f(H/H_{0} )]F_{0} (H/H_{0} )}}} \right)^{{m(H/H_{0} )}} } \right.} \right] \\ & \quad + 2\mu f(H/H_{0} ) \\ \end{aligned}$$
(24)

The model comprehensively considers the occurrence depth of this coal, can simulate the entire process of dynamic deformation and failure of coal at different depths, and reflects the nonlinear relationship between dynamic stress and dynamic strain in dynamic deformation of coal. Therefore, the established damage constitutive model can effectively describe the dynamic deformation mechanical mechanism of coal at different depths.

3.2 Model verification

The established constitutive models were verified based on established dynamic damage constitutive models of coal at different depths combined with test data. According to dynamic stress–strain curves of coal at different depths under different loading rates, parameters corresponding to loading rates at different depths were obtained using Eqs. (16), (17), and (20), as shown in Table 4. Substituting the test parameters in Table 4 into Eq. (24), the theoretical value of the dynamic stress–strain curve of coal at different depths under different loading rates is compared with the test curve, as shown in Fig. 8.

Table 4 Parameters for different depths corresponding to loading rates
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Fig. 8
figure 8

Comparison between theoretical and experimental curves of dynamic stress–strain curves of coal at different depths under different loading rates

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Given the lack of key parameters collected after the peak during the trial, the theoretical and experimental curves obtained by the established coal constitutive model at different depths are in good agreement below the peak stress, which indicates that the established model has good applicability to the dynamic mechanical properties of coal at different depths in this stress range, and verifies the validity of the proposed coal dynamic constitutive model for different depths.

3.3 Influence of model parameters of coal at different depths

Poisson’s ratio is difficult to measure in dynamic mechanical tests, and thus the static Poisson’s ratio was adopted in this model. Mechanical parameters obtained by static uniaxial compression testing of three homogeneous samples are shown in Table 5.

Table 5 Mechanical parameters for uniaxial compression failure testing of coal
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The influence of Poisson’s ratio on the dynamic stress–strain curve of coal is discussed based on the above experiments, and taking the dynamic stress–strain curve of coal at a depth of 400 m and loading rate of 517 GPa/s as an example. The values of model parameters m and F0, corresponding to Poisson’s ratio between 0.20 and 0.45, are shown in Table 6. The constitutive model parameter m gradually decreases with increase of Poisson’s ratio, and F0 is positively correlated with Poisson’s ratio, but the overall change is small. As shown in Table 6, compared with the static Poisson’s ratio of 0.32, m increased by 0.39%, and F0 decreased by 0.79% when Poisson’s ratio was 0.20; when Poisson’s ratio was 0.45, m decreased by 0.47%, and F0 increased by 0.95%. The range of parameters m and F0 varied by less than 1%, and the error is considered acceptable.

Table 6 Variation of model parameters m and F0 with Poisson’s ratio
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Matlab programming was employed to introduce different Poisson’s ratios into the proposed constitutive model and the dynamic stress–strain curves of coal corresponding to different Poisson’s ratios, as shown in Fig. 9. Poisson’s ratio has little effect on the dynamic stress–strain curves of coal, which is in good agreement with the experimental dynamic stress–strain curves, indicating that values of Poisson’s ratio obtained by static tests can be used to calculate the proposed constitutive model.

Fig. 9
figure 9

Dynamic stress–strain curves of coal corresponding to different Poisson ratios

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4 Conclusion

Considering coal in Pingdingshan mine at a depth of 1000 m as the research object, dynamic mechanical tests of the coal at different depths were carried out using an improved SHPB experimental system. Relevant mechanical calculation formulas were derived based on the proposed dynamic strength criteria of coal at different depths, and the infinitesimal element strength of coal was determined. Deviation between the calculation using the static Poisson’s ratio and the test results was compared and a dynamic constitutive model of coal at different depths was constructed. The theoretical model was verified by dynamic stress–strain curves of coal under different depth stress conditions.

  1. 1.

    With the increase of occurrence depth, the dynamic peak stress of coal increases overall, with the maximum increase of 66.62%, while the peak strain decreases, with the minimum decrease of 19.30%.

  2. 2.

    When the depth of occurrence is the same, the dynamic elastic modulus of coal is less affected by the loading rate. As the depth of occurrence increases, the dynamic elastic modulus of coal shows a significant increase trend.

  3. 3.

    An expression for coal infinitesimal element strength that considers the rate effect was determined. The proposed method for measuring the dynamic infinitesimal element strength of coal is universal according to the meanings of the expression parameters.

  4. 4.

    A dynamic damage constitutive model of coal that considers occurrence depth and the parameters for different occurrence depths were determined. The model was verified, which showed that it has universality for coal at different depths prior to peak stress.

  5. 5.

    The static Poisson’s ratio can be used as the determination parameter of the coal dynamic constitutive model, and the influence error is less than 1%.